(* Advanced inversion lemmas ************************************************)
-lemma scpes_inv_abst2: ∀h,g,a,G,L,T1,T2,W2,l1,l2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l2] ⓛ{a}W2.T2 →
- ∃∃W,T. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, l1] ⓛ{a}W.T & ⦃G, L⦄ ⊢ W2 ➡* W &
- ⦃G, L.ⓛW2⦄ ⊢ T2 •*➡*[h, g, l2] T.
-#h #g #a #G #L #T1 #T2 #W2 #l1 #l2 * #T0 #HT10 #H
+lemma scpes_inv_abst2: ∀h,o,a,G,L,T1,T2,W2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] ⓛ{a}W2.T2 →
+ ∃∃W,T. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] ⓛ{a}W.T & ⦃G, L⦄ ⊢ W2 ➡* W &
+ ⦃G, L.ⓛW2⦄ ⊢ T2 •*➡*[h, o, d2] T.
+#h #o #a #G #L #T1 #T2 #W2 #d1 #d2 * #T0 #HT10 #H
elim (scpds_inv_abst1 … H) -H #W #T #HW2 #HT2 #H destruct /2 width=5 by ex3_2_intro/
qed-.
(* Advanced properties ******************************************************)
-lemma scpes_refl: ∀h,g,G,L,T,l1,l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ T ▪[h, g] l1 →
- ⦃G, L⦄ ⊢ T •*⬌*[h, g, l2, l2] T.
-#h #g #G #L #T #l1 #l2 #Hl21 #Hl1
-elim (da_lstas … Hl1 … l2) #U #HTU #_
+lemma scpes_refl: ∀h,o,G,L,T,d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
+ ⦃G, L⦄ ⊢ T •*⬌*[h, o, d2, d2] T.
+#h #o #G #L #T #d1 #d2 #Hd21 #Hd1
+elim (da_lstas … Hd1 … d2) #U #HTU #_
/3 width=3 by scpds_div, lstas_scpds/
qed.
-lemma lstas_scpes_trans: ∀h,g,G,L,T1,l0,l1. ⦃G, L⦄ ⊢ T1 ▪[h, g] l0 → l1 ≤ l0 →
- ∀T. ⦃G, L⦄ ⊢ T1 •*[h, l1] T →
- ∀T2,l,l2. ⦃G, L⦄ ⊢ T •*⬌*[h,g,l,l2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h,g,l1+l,l2] T2.
-#h #g #G #L #T1 #l0 #l1 #Hl0 #Hl10 #T #HT1 #T2 #l #l2 *
+lemma lstas_scpes_trans: ∀h,o,G,L,T1,d0,d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d0 → d1 ≤ d0 →
+ ∀T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
+ ∀T2,d,d2. ⦃G, L⦄ ⊢ T •*⬌*[h,o,d,d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h,o,d1+d,d2] T2.
+#h #o #G #L #T1 #d0 #d1 #Hd0 #Hd10 #T #HT1 #T2 #d #d2 *
/3 width=3 by scpds_div, lstas_scpds_trans/ qed-.
(* Properties on parallel computation for terms *****************************)
-lemma cprs_scpds_div: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T →
- ∀l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l →
- ∀T2,l2. ⦃G, L⦄ ⊢ T2 •*➡*[h, g, l2] T →
- ⦃G, L⦄⊢ T1 •*⬌*[h, g, 0, l2] T2.
-#h #g #G #L #T1 #T #HT1 #l #Hl elim (da_lstas … Hl 0)
+lemma cprs_scpds_div: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T →
+ ∀d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T →
+ ⦃G, L⦄⊢ T1 •*⬌*[h, o, 0, d2] T2.
+#h #o #G #L #T1 #T #HT1 #d #Hd elim (da_lstas … Hd 0)
#X1 #HTX1 #_ elim (cprs_strip … HT1 X1) -HT1
/3 width=5 by scpds_strap1, scpds_div, lstas_cpr, ex4_2_intro/
qed.
(* Main properties **********************************************************)
-theorem scpes_trans: ∀h,g,G,L,T1,T,l1,l. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l] T →
- ∀T2,l2. ⦃G, L⦄ ⊢ T •*⬌*[h, g, l, l2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2.
-#h #g #G #L #T1 #T #l1 #l * #X1 #HT1X1 #HTX1 #T2 #l2 * #X2 #HTX2 #HT2X2
-elim (scpds_conf_eq … HTX1 … HTX2) -T -l /3 width=5 by scpds_cprs_trans, scpds_div/
+theorem scpes_trans: ∀h,o,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d] T →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
+#h #o #G #L #T1 #T #d1 #d * #X1 #HT1X1 #HTX1 #T2 #d2 * #X2 #HTX2 #HT2X2
+elim (scpds_conf_eq … HTX1 … HTX2) -T -d /3 width=5 by scpds_cprs_trans, scpds_div/
qed-.
-theorem scpes_canc_sn: ∀h,g,G,L,T,T1,l,l1. ⦃G, L⦄ ⊢ T •*⬌*[h, g, l, l1] T1 →
- ∀T2,l2. ⦃G, L⦄ ⊢ T •*⬌*[h, g, l, l2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2.
+theorem scpes_canc_sn: ∀h,o,G,L,T,T1,d,d1. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d1] T1 →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T •*⬌*[h, o, d, d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
/3 width=4 by scpes_trans, scpes_sym/ qed-.
-theorem scpes_canc_dx: ∀h,g,G,L,T1,T,l1,l. ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l] T →
- ∀T2,l2. ⦃G, L⦄ ⊢ T2 •*⬌*[h, g, l2, l] T → ⦃G, L⦄ ⊢ T1 •*⬌*[h, g, l1, l2] T2.
+theorem scpes_canc_dx: ∀h,o,G,L,T1,T,d1,d. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d] T →
+ ∀T2,d2. ⦃G, L⦄ ⊢ T2 •*⬌*[h, o, d2, d] T → ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
/3 width=4 by scpes_trans, scpes_sym/ qed-.